Fix a complex structure \(J\)  on \(S\)  and denote by \(X\)  the Riemann surface \((S,J)\) . Then for any hyperbolic metric \(h \in \text{Teich}(S)\)  there is [1]} a unique [2]} harmonic map \(w_h \colon X \rightarrow (S,h)\)  in the homotopy class of the identity. Harmonicity of \(w_h\)  ensures that the Hopf differential \(q_h=\left(w_h^{*}h\right)^{(2,0)}\)  is a holomorphic quadratic differential on \(X\) . The vector space \(\text{QD}(X)\)  of holomorphic quadratic differentials on \(X\)  has a natural norm given by the \(L^{2}\) -norm with respect to the uniformizing hyperbolic metric \(\sigma \)  of \(X\) . With an abuse of notation, we will still denote by \(X\)  the hyperbolic surface \((S,\sigma )\) . The map which assigns to a point in Teichmüller space its corresponding Hopf differential is a homeomorphism [3]}.
